Abstract

A method to realize effective magnetic fields for photons in square lattices of coupled optical waveguides or resonators is suggested, which is inspired by an optical analogue of photon-assisted tunneling of atom optics. It is shown that an artificial magnetic field can be achieved by application of an index gradient and periodic lumped phase shifts or modulation of the propagation constants/resonances, without the need to modulate the coupling strength.

© 2013 Optical Society of America

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  1. S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
    [CrossRef]
  2. F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
    [CrossRef]
  3. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
    [CrossRef]
  4. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772775 (2009).
  5. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
    [CrossRef]
  6. R. O. Umucallar and I. Carusotto, Phys. Rev. A 84, 043804 (2011).
    [CrossRef]
  7. K. Fang, Z. Yu, and S. Fan, Phys. Rev. Lett. 108, 153901 (2012).
    [CrossRef]
  8. K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
    [CrossRef]
  9. M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).
  10. L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
    [CrossRef]
  11. D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
    [CrossRef]
  12. C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
    [CrossRef]
  13. A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
    [CrossRef]
  14. S. Longhi, Phys. Rev. B 76, 19511 (2007).
  15. D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003).
    [CrossRef]
  16. A. R. Kolovsky, Europhys Lett. 93, 20003 (2011).
  17. M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
    [CrossRef]
  18. A. Bermudez, T. Schaetz, and D. Porras, New J. Phys. 14, 053049 (2012).
    [CrossRef]
  19. H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
    [CrossRef]
  20. S. Aubry and G. André, Ann. Isr. Phys. Soc. 3, 133 (1980).
  21. A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
    [CrossRef]
  22. S. Longhi, Laser Photon. Rev. 3, 243 (2009).
    [CrossRef]
  23. A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
    [CrossRef]
  24. A. R. Kolovsky and G. Mantica, Phys. Rev. B 86, 054306 (2012).
    [CrossRef]

2013

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
[CrossRef]

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

2012

A. Bermudez, T. Schaetz, and D. Porras, New J. Phys. 14, 053049 (2012).
[CrossRef]

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Phys. Rev. Lett. 108, 153901 (2012).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

A. R. Kolovsky and G. Mantica, Phys. Rev. B 86, 054306 (2012).
[CrossRef]

2011

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

R. O. Umucallar and I. Carusotto, Phys. Rev. A 84, 043804 (2011).
[CrossRef]

A. R. Kolovsky, Europhys Lett. 93, 20003 (2011).

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

2009

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

S. Longhi, Laser Photon. Rev. 3, 243 (2009).
[CrossRef]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772775 (2009).

2008

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

2007

S. Longhi, Phys. Rev. B 76, 19511 (2007).

2003

D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003).
[CrossRef]

1980

S. Aubry and G. André, Ann. Isr. Phys. Soc. 3, 133 (1980).

1976

D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
[CrossRef]

Aidelsburger, M.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

André, G.

S. Aubry and G. André, Ann. Isr. Phys. Soc. 3, 133 (1980).

Atala, M.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Aubry, S.

S. Aubry and G. André, Ann. Isr. Phys. Soc. 3, 133 (1980).

Barthelemy, A.

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Bermudez, A.

A. Bermudez, T. Schaetz, and D. Porras, New J. Phys. 14, 053049 (2012).
[CrossRef]

Bloch, I.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Carusotto, I.

R. O. Umucallar and I. Carusotto, Phys. Rev. A 84, 043804 (2011).
[CrossRef]

Chen, Y.-A.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772775 (2009).

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Dean, C. R.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Dreisow, F.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Fan, S.

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Phys. Rev. Lett. 108, 153901 (2012).
[CrossRef]

Fang, K.

K. Fang, Z. Yu, and S. Fan, Phys. Rev. Lett. 108, 153901 (2012).
[CrossRef]

K. Fang, Z. Yu, and S. Fan, Nat. Photonics 6, 782 (2012).
[CrossRef]

Forsythe, C.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Fu, L.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
[CrossRef]

Gao, Y.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Ghahari, F.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Hafezi, M.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Heinrich, M.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Hofstadter, D. R.

D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).
[CrossRef]

Hone, J.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Hossein Mousavi, S.

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

Ishigami, M.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Jaksch, D.

D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003).
[CrossRef]

Joannopoulos, J. D.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
[CrossRef]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772775 (2009).

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Kargarian, M.

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

Kartashov, Y. V.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

Katoch, J.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Khanikaev, A. B.

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

Kim, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Kolovsky, A. R.

A. R. Kolovsky and G. Mantica, Phys. Rev. B 86, 054306 (2012).
[CrossRef]

A. R. Kolovsky, Europhys Lett. 93, 20003 (2011).

Koshino, M.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Lederer, F.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

Lipson, M.

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[CrossRef]

Lira, H.

H. Lira, Z. Yu, S. Fan, and M. Lipson, Phys. Rev. Lett. 109, 033901 (2012).
[CrossRef]

Longhi, S.

S. Longhi, Laser Photon. Rev. 3, 243 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

S. Longhi, Phys. Rev. B 76, 19511 (2007).

Louradour, F.

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Lu, L.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
[CrossRef]

Lukin, M. D.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

MacDonald, A. H.

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

Maher, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Mantica, G.

A. R. Kolovsky and G. Mantica, Phys. Rev. B 86, 054306 (2012).
[CrossRef]

Moon, P.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Nascimbene, S.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Nolte, S.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Pertsch, T.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Porras, D.

A. Bermudez, T. Schaetz, and D. Porras, New J. Phys. 14, 053049 (2012).
[CrossRef]

Raghu, S.

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

Rechtsman, M. C.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).

Schaetz, T.

A. Bermudez, T. Schaetz, and D. Porras, New J. Phys. 14, 053049 (2012).
[CrossRef]

Segev, M.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).

Shepard, K. L.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Shvets, G.

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, Nat. Mater. 12, 233 (2013).
[CrossRef]

Soljacic, M.

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, Nat. Photonics 7, 294 (2013).
[CrossRef]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Nature 461, 772775 (2009).

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacic, Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef]

Suran, E.

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Szameit, A.

M. C. Rechtsman, J. M. Zeuner, A. Tünnermann, S. Nolte, M. Segev, and A. Szameit, Nat. Photonics 7, 153158 (2013).

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

A. Szameit, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, E. Suran, F. Louradour, A. Barthelemy, and S. Longhi, Appl. Phys. Lett. 93, 181109 (2008).
[CrossRef]

Taniguchi, T.

C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and P. Kim, “Hofstadters butterfly and the fractal quantum Hall effect in moir superlattices,” Nature (to be published), doi: 10.1038/nature12186.
[CrossRef]

Taylor, J. M.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, Nat. Phys. 7, 907 (2011).
[CrossRef]

Torner, L.

A. Szameit, Y. V. Kartashov, F. Dreisow, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, V. A. Vysloukh, F. Lederer, and L. Torner, Phys. Rev. Lett. 102, 153901 (2009).
[CrossRef]

Trotzky, S.

M. Aidelsburger, M. Atala, S. Nascimbene, S. Trotzky, Y.-A. Chen, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011).
[CrossRef]

Tse, W.-K.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Schematic of a square lattice of optical waveguides/resonators. (b) Interference fringe [evolution of In(t)=m|cn,m(t)|2] in a square lattice of waveguides illuminated by a broad Gaussian beam for a rational flux number α=1/2 as obtained by numerical simulations of Eq. (1) (upper panel) and from the effective Eq. (8) (lower panel). (c) Same as (b), but for an irrational flux number α=3/(2π). Parameter values are given in the text.

Fig. 2.
Fig. 2.

Cyclotron motion of an initial Gaussian wave packet [path described by the beam center of mass (n(t), m(t)] as obtained (a) from numerical simulations of Eq. (1), (b) from the effective Eq. (8), and (c) from the semi-classical analysis. In (d), snapshots of the beam intensity distribution |cn,m(t)|2 at a few normalized propagation distances Jxt are shown. Parameter values are given in the text.

Equations (16)

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idcn,mdt=Jx(cn+1,m+cn1,m)Jy(cn,m+1+cn,m1)+βn,m(t)cn,m,
βn,m(t)=β0+Fm+Vn,m(t),
F=Mω
Vn,m(t)=AH(ωt+ϕn,m),
ϕn,m=nσ+mρ,
cn,m(t)=fn,m(t)×exp[iφn,mi0tdtβn,m(t)]
φn,m=G(ϕn,m)+M2ρm(m1),
idfn,mdt=κxfn+1,mκx*fn1,mκyexp(inMσ)fn,m+1κy*exp(inMσ)fn,m1,
κx=Jx2π02πdxexp{iΓ[G(x)G(x+σ)]},
κy=Jy2π02πdxexp{iMx+iΓ[G(x)G(x+ρ)]},
κx=JxJ0(2Γsin(σ2)),
κy=JyJM(2Γsin(ρ2))exp[iM(ρπ)/2].
κx=Jx(12|σ|πsin2(Γ2)),
κy=4JyMπsin(Mρ2)sin(Γ2)sin(Mπ2ρ|ρ|Γ2)×exp[iM(ρπ)/2].
dndt=2κxsinP^n,dmdt=2κysinP^m,
dP^ndt=2κyσsinP^m,dP^mdt=2κxσsinP^n,

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